The CBS News program “60 Minutes” was moving forward with plans to show an interview it conducted last week with the pornographic film actress who says she had an affair with Donald J. Trump as her lawyer and a lawyer for the president traded public jabs through the weekend over her right to speak.

The porn star, Stephanie Clifford, spoke with the “60 Minutes” contributor Anderson Cooper late last week. She did so despite an arbitrator’s ruling reaffirming an agreement she reached with Mr. Trump in October 2016 to remain silent about their alleged relationship — which she said started in 2006 and lasted several months — in exchange for $130,000.

The “60 Minutes” interview raised the prospect that Mr. Trump’s personal attorney and fixer, Michael D. Cohen, would seek an injunction stopping CBS from airing the segment, which does not yet have a scheduled broadcast date. But as of Sunday night CBS had not received any legal threat. Mr. Cohen did not immediately respond to questions on Sunday.

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With the “60 Minutes” interview and a series of tweets from Ms. Clifford’s lawyer, Michael Avenatti, taunting Mr. Cohen last week, the actress and her lawyer were pursuing an aggressive strategy to pressure the president. Their goal: releasing Ms. Clifford, who goes by the stage name Stormy Daniels, from her agreement.

“Our aim and our messaging is very simple: We’re going to shoot straight, we’re going to provide evidence and facts, and we are going to consistently advocate for the American people being able to make their own decisions as to who’s telling the truth and who’s lying to them,” Mr. Avenatti said. “She wants a forum to tell her version of events and let the chips fall where they may.”

The campaign was not without its risks, legal and otherwise. Mr. Avenatti said he was assigning 24-hour security to Ms. Clifford after a car tailed her after a dance performance at a Florida strip club this weekend.

He first hinted that there was a “60 Minutes” interview himself on Thursday, when he posted a tweet featuring a photograph of him, Ms. Clifford and Mr. Cooper without comment.

Ms. Clifford hired Mr. Avenatti a couple of weeks ago. (She was previously represented by another California attorney, Keith Davidson.) Within days Mr. Avenatti prepared to challenge the deal — which stipulated that disagreements would be argued in secret before a confidential arbitrator — arguing it was not valid because Mr. Trump did not sign it. (Mr. Avenatti on Sunday pointed to a clause that said the agreement was “valid and binding” when “signed by all parties.”)

A suit that Mr. Avenatti filed on Tuesday evening in Los Angeles County Superior Court included a nugget of news: Mr. Trump’s lawyer had initiated “a bogus arbitration proceeding” against Ms. Clifford to “shut her up.”

That appeared to force an error at the White House the next day, when the president’s press secretary, Sarah Huckabee Sanders, who had been avoiding engagement in Ms. Clifford’s story, said that an arbitration proceeding “was won in the president’s favor.” That led to news that Mr. Cohen had acquired a temporary restraining order against Ms. Clifford in late February.

Mr. Avenatti has been a regular presence on television, arguing that the restraining order and the original deal are both legally flawed, and making allegations aimed at undermining Mr. Cohen and Mr. Trump.

For instance, on Friday, Mr. Avenatti shared with NBC News and then ABC News that Mr. Cohen had conducted some negotiations with Ms. Clifford in 2016 on his Trump Organization email account. That, Mr. Avenatti told NBC, was at variance with Mr. Cohen’s previous statement that neither the Trump Organization nor the Trump campaign were “party to” the agreement with Ms. Clifford. Mr. Avenatti also suggested that the email cast doubt on Mr. Cohen’s assertion that he paid Ms. Clifford out of his own pocket.

Whether Mr. Trump ever reimbursed Mr. Cohen, and the extent to which he struck the deal in consultation with Mr. Trump, goes to the center of complaints the watchdog group Common Cause has made to the Federal Election Commission and the Justice Department alleging that the payment was an illegal campaign contribution.

After Mr. Avenatti’s comments, Mr. Cohen broke his relative silence about the case by telling ABC News that he drew on his own home equity line of credit to secure the $130,000 for Ms. Clifford. And The Washington Post quoted him over the weekend as saying, “Mr. Avenatti’s actions and behavior has been both reckless and imprudent as it opens Ms. Clifford to substantial monetary liability, which I intend to pursue.”

Mr. Avenatti answered with a Twitter message on Saturday taunting Mr. Cohen by asking on whose behalf he would pursue the financial penalty, given the absence of Mr. Trump’s signature on the October contract.

This time, Mr. Cohen did not respond.

But Mr. Cohen’s most important next moves may be his response to Ms. Clifford’s suit, as well as any action he may take to try to block “60 Minutes,” which could create a major First Amendment standoff. CBS had no comment on Sunday.

A person familiar with the network’s preparations said that “60 Minutes” had not received notice of a legal action in relation to the interview with Ms. Clifford. Producers at “60 Minutes” are preparing the segment, a process that often includes a legal review and fact-checking.

Maggie Haberman contributed reporting.

A version of this article appears in print on , on Page A13 of the New York edition with the headline: Lawyers Wrangle as ‘60 Minutes’ Prepares to Air Porn Star Interview. Order Reprints | Today’s Paper | Subscribe